Standard Bases in mixed Power Series and Polynomial Rings over Rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring $R[[t_1,\dots ,t_m]][x_1,\dots ,x_n{]}^s$ respectively of their localization with respect to a $t$-local monomial ordering for a certain class of noetherian rings $R$. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert–Hironaka and Mora and to generalize the Buchberger criterion. Everything else then translates naturally. Setting either $m=0$ or $n=0$ we get standard bases for polynomial rings respectively for power series rings over $R$ as a special case.